Table of flat knot invariants
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Flat knot 5.106

Min(phi) over symmetries of the knot is: [-1,-1,1,1,0,0,1,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['4.10', '5.106', '6.2037']
Arrow polynomial of the knot is: 4*K1**2 - 2*K2 - 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.24', '5.34', '5.47', '5.56', '5.62', '5.66', '5.67', '5.75', '5.89', '5.92', '5.94', '5.97', '5.102', '5.103', '5.105', '5.106', '5.110', '5.111', '5.115', '5.116', '5.117', '5.118', '5.119', '7.956', '7.5918', '7.8159', '7.9694', '7.10242', '7.10377', '7.10426', '7.10662', '7.11274', '7.11712', '7.12250', '7.12276', '7.13486', '7.14878', '7.15090', '7.16067', '7.16079', '7.16119', '7.16509', '7.16510', '7.17050', '7.17134', '7.17538', '7.17747', '7.18421', '7.18428', '7.18432', '7.18464', '7.18468', '7.18484', '7.18494', '7.18533', '7.18575', '7.18585', '7.18806', '7.18972', '7.18994', '7.18996', '7.19006', '7.19028', '7.19089', '7.19116', '7.19688', '7.19953', '7.20100', '7.20218', '7.20318', '7.20429', '7.20452', '7.21133', '7.21134', '7.21144', '7.21237', '7.21238', '7.21353', '7.21357', '7.21361', '7.21373', '7.21442', '7.21445', '7.21451', '7.21458', '7.21570', '7.21727', '7.21792', '7.22130', '7.22840', '7.22880', '7.23312', '7.23724', '7.23849', '7.24143', '7.24174', '7.24191', '7.24286', '7.24309', '7.24344', '7.24643', '7.24695', '7.24702', '7.24724', '7.24735', '7.24781', '7.24863', '7.24907', '7.24915', '7.24949', '7.24991', '7.25096', '7.25465', '7.25842', '7.25843', '7.25847', '7.25872', '7.25889', '7.25895', '7.25953', '7.25968', '7.25992', '7.25996', '7.25997', '7.26004', '7.26019', '7.26078', '7.26107', '7.26440', '7.26662', '7.26797', '7.26801', '7.26816', '7.27009', '7.27022', '7.27028', '7.27091', '7.27207', '7.27299', '7.27303', '7.27506', '7.27510', '7.27512', '7.27540', '7.27543', '7.27547', '7.27548', '7.27552', '7.27578', '7.27583', '7.27601', '7.27615', '7.27675', '7.27690', '7.27698', '7.27742', '7.27748', '7.27761', '7.27764', '7.27768', '7.27813', '7.27827', '7.27845', '7.27849', '7.27853', '7.27867', '7.27869', '7.27879', '7.27888', '7.27930', '7.28175', '7.28176', '7.28193', '7.28201', '7.28212', '7.28378', '7.28390', '7.28397', '7.28490', '7.28506', '7.28625', '7.28684', '7.28711', '7.28716', '7.28727', '7.28890', '7.28983', '7.29002', '7.29015', '7.29056', '7.29060', '7.29071', '7.29114', '7.29461', '7.29483', '7.29681', '7.29722', '7.30162', '7.30175', '7.30406', '7.30503', '7.30679', '7.30969', '7.31035', '7.31092', '7.31133', '7.31259', '7.31309', '7.31344', '7.31374', '7.31427', '7.31515', '7.31564', '7.31582', '7.31631', '7.31668', '7.31670', '7.31676', '7.31702', '7.31723', '7.31726', '7.31754', '7.31780', '7.31837', '7.32049', '7.32075', '7.32076', '7.32153', '7.32208', '7.32222', '7.32508', '7.32512', '7.32513', '7.32518', '7.32543', '7.32555', '7.32584', '7.32585', '7.32591', '7.32601', '7.32606', '7.32633', '7.32684', '7.32687', '7.32704', '7.32732', '7.32761', '7.32770', '7.32775', '7.32791', '7.32825', '7.32827', '7.32832', '7.32851', '7.32890', '7.32897', '7.32902', '7.32913', '7.32941', '7.32970', '7.32996', '7.32998', '7.33042', '7.33045', '7.33070', '7.33081', '7.33110', '7.33118', '7.33146', '7.33174', '7.33187', '7.33197', '7.33219', '7.33221', '7.33229', '7.33237', '7.33250', '7.33280', '7.33294', '7.33308', '7.33361', '7.33391', '7.33516', '7.33550', '7.33553', '7.33578', '7.33700', '7.33721', '7.33743', '7.33814', '7.33830', '7.33832', '7.33843', '7.33977', '7.33981', '7.34051', '7.34148', '7.34175', '7.34180', '7.34182', '7.34206', '7.34272', '7.34290', '7.34291', '7.34336', '7.34347', '7.34436', '7.34492', '7.34496', '7.34524', '7.34535', '7.34546', '7.34553', '7.34596', '7.34601', '7.34638', '7.34682', '7.34683', '7.34684', '7.34698', '7.34714', '7.34717', '7.34782', '7.34784', '7.34785', '7.34818', '7.34821', '7.34850', '7.34876', '7.34949', '7.34951', '7.34978', '7.34983', '7.34984', '7.34986', '7.35030', '7.35040', '7.35041', '7.35048', '7.35063', '7.35071', '7.35128', '7.35129', '7.35135', '7.35149', '7.35159', '7.35202', '7.35208', '7.35211', '7.35213', '7.35217', '7.35228', '7.35230', '7.35244', '7.35253', '7.35260', '7.35269', '7.35301', '7.35304', '7.35307', '7.35322', '7.35325', '7.35331', '7.35336', '7.35349', '7.35362', '7.35363', '7.35371', '7.35372', '7.35375', '7.35376', '7.35377', '7.35380', '7.35381', '7.35382', '7.35383', '7.35384', '7.35385', '7.35386', '7.35387', '7.35401', '7.35403', '7.35425', '7.35433', '7.35444', '7.35445', '7.35446', '7.35447', '7.35448', '7.35449', '7.35463', '7.35466', '7.35474', '7.35476', '7.35488', '7.35489', '7.35490', '7.35491', '7.35493', '7.35505', '7.35511', '7.35521', '7.35523', '7.35531', '7.35536', '7.35537', '7.35542', '7.35543', '7.35545', '7.35546', '7.35547', '7.35551', '7.35557', '7.35561', '7.35574', '7.35582', '7.35583', '7.35586', '7.35587', '7.35606', '7.35634', '7.35636', '7.35661', '7.35662', '7.35670', '7.35675', '7.35676', '7.35730', '7.35926', '7.35986', '7.36068', '7.36320', '7.36322', '7.36331', '7.36336', '7.36397', '7.36400', '7.36428', '7.36441', '7.36466', '7.36483', '7.36510', '7.36527', '7.36561', '7.36564', '7.36570', '7.36573', '7.36574', '7.36576', '7.36579', '7.36605', '7.36610', '7.36613', '7.36622', 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'7.39181', '7.39207', '7.39215', '7.39217', '7.39235', '7.39242', '7.39249', '7.39250', '7.39309', '7.39330', '7.39342', '7.39344', '7.39350', '7.39352', '7.39482', '7.39523', '7.39530', '7.39546', '7.39556', '7.39586', '7.39639', '7.39642', '7.39646', '7.39656', '7.39707', '7.39708', '7.39711', '7.39717', '7.39759', '7.39785', '7.40329', '7.40392', '7.40424', '7.40425', '7.40426', '7.40427', '7.40429', '7.40431', '7.40453', '7.40462', '7.40463', '7.40465', '7.40466', '7.40468', '7.40510', '7.40542', '7.40543', '7.40548', '7.40563', '7.40640', '7.40641', '7.40666', '7.40785', '7.40819', '7.40829', '7.40865', '7.40886', '7.40902', '7.40940', '7.40986', '7.40999', '7.41001', '7.41062', '7.41078', '7.41301', '7.41302', '7.41325', '7.41410', '7.41451', '7.41455', '7.41464', '7.41604', '7.41667', '7.41707', '7.41737', '7.41749', '7.41885', '7.41886', '7.41900', '7.41946', '7.42021', '7.42069', '7.42076', '7.42077', '7.42078', '7.42087', '7.42136', '7.42147', '7.42157', '7.42167', '7.42168', '7.42169', '7.42178', '7.42181', '7.42182', '7.42200', '7.42252', '7.42270', '7.42271', '7.42276', '7.42277', '7.42281', '7.42288', '7.42389', '7.42398', '7.42447', '7.42536', '7.42595', '7.42601', '7.42649', '7.42684', '7.42685', '7.42720', '7.42836', '7.42842', '7.42844', '7.42868', '7.42872', '7.42873', '7.42880', '7.42905', '7.42906', '7.42911', '7.42912', '7.42931', '7.42938', '7.42960', '7.42973', '7.42994', '7.43006', '7.43010', '7.43011', '7.43012', '7.43013', '7.43021', '7.43024', '7.43028', '7.43044', '7.43060', '7.43099', '7.43102', '7.43111', '7.43112', '7.43114', '7.43120', '7.43131', '7.43134', '7.43136', '7.43147', '7.43151', '7.43166', '7.43172', '7.43186', '7.43187', '7.43189', '7.43200', '7.43203', '7.43215', '7.43240', '7.43253', '7.43278', '7.43403', '7.43405', '7.43406', '7.43411', '7.43432', '7.43445', '7.43455', '7.43465', '7.43482', '7.43486', '7.43524', '7.43527', '7.43534', '7.43536', '7.43537', '7.43574', '7.43578', '7.43579', '7.43582', '7.43590', '7.43593', 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'7.44603', '7.44604', '7.44605', '7.44613', '7.44614', '7.44616', '7.44617', '7.44618', '7.44626', '7.44628', '7.44629', '7.44630', '7.44631', '7.44632', '7.44634', '7.44635', '7.44638', '7.44639', '7.44640', '7.44644', '7.44645', '7.44651', '7.44652', '7.44653', '7.44657', '7.44659', '7.44662', '7.44709', '7.44710', '7.44733', '7.44734', '7.44792', '7.44847', '7.44980', '7.44994', '7.44997', '7.44998', '7.45004', '7.45008', '7.45011', '7.45023', '7.45024', '7.45025', '7.45029', '7.45042', '7.45079', '7.45087', '7.45088', '7.45090', '7.45093', '7.45100', '7.45104', '7.45109', '7.45110', '7.45112', '7.45121', '7.45125', '7.45126', '7.45127', '7.45128', '7.45129', '7.45135', '7.45136', '7.45140', '7.45142', '7.45144', '7.45152', '7.45157', '7.45160', '7.45170', '7.45183', '7.45184', '7.45187', '7.45292', '7.45297', '7.45314', '7.45316', '7.45325', '7.45326', '7.45374', '7.45382', '7.45385', '7.45392', '7.45467', '7.45471', '7.45472', '7.45473', '7.45480', '7.45483', '7.45485', '7.45486', '7.45488', '7.45496', '7.45501', '7.45502', '7.45503', '7.45505', '7.45506', '7.45508', '7.45510', '7.45511', '7.45512', '7.45513', '7.45514', '7.45515', '7.45520', '7.45521', '7.45525', '7.45526', '7.45528', '7.45529', '7.45530', '7.45531', '7.45532', '7.45533', '7.45534', '7.45540', '7.45545', '7.45549', '7.45553', '7.45557', '7.45558', '7.45559', '7.45567', '7.45569', '7.45578', '7.45581', '7.45589', '7.45598', '7.45603', '7.45609', '7.45611', '7.45616', '7.45617', '7.45620', '7.45623', '7.45624', '7.45625', '7.45635', '7.45636', '7.45638', '7.45641', '7.45642', '7.45648', '7.45649', '7.45652', '7.45653', '7.45655', '7.45656', '7.45658', '7.45659', '7.45660', '7.45663', '7.45667', '7.45670', '7.45674', '7.45676', '7.45681', '7.45682', '7.45684', '7.45686', '7.45687', '7.45693', '7.45694', '7.45698', '7.45713', '7.45720', '7.45722', '7.45723', '7.45724', '7.45731', '7.45732', '7.45737', '7.45746', '7.45747', '7.45752', '7.45753', '7.45756', '7.45761', '7.45766', '7.45767', '7.45780', '7.45783', '7.45786', '7.45790', '7.45791', '7.45800', '7.45801', '7.45806', '7.45810', '7.45818', '7.45823', '7.45824', '7.45825', '7.45826', '7.45829', '7.45831', '7.45832', '7.45833', '7.45834', '7.45835', '7.45836', '7.45838', '7.45844', '7.45845', '7.45846', '7.45850', '7.45854', '7.45855', '7.45856', '7.45857', '7.45858', '7.45859', '7.45860', '7.45863', '7.45865', '7.45866', '7.45867', '7.45871', '7.45872', '7.45873', '7.45874', '7.45879', '7.45880', '7.45883', '7.45884', '7.45886', '7.45887', '7.45889', '7.45891', '7.45892', '7.45893', '7.45897', '7.45898', '7.45899', '7.45900', '7.45902', '7.45903', '7.45906', '7.45907', '7.45908', '7.45909', '7.45913', '7.45918', '7.45921', '7.45925', '7.45949', '7.45960', '7.45961', '7.45967', '7.46006', '7.46031', '7.46033', '7.46051', '7.46059', '7.46063', '7.46067', '7.46069', '7.46070', '7.46073', '7.46110', '7.46118', '7.46119', '7.46125', '7.46134', '7.46140', '7.46149', '7.46187', '7.46188', '7.46192', '7.46205', '7.46218']
Outer characteristic polynomial of the knot is: t^5+6t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.10', '5.106', '6.2037']
2-strand cable arrow polynomial of the knot is: -1536*K1**4 - 1280*K1**2*K2**2 + 1728*K1**2*K2 + 400*K1**2 + 736*K1*K2*K3 - 176*K2**4 + 96*K2**2*K4 - 128*K2**2 - 48*K3**2 - 4*K4**2 + 210
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.106']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.3', 'vk5.10', 'vk5.43', 'vk5.50', 'vk5.93', 'vk5.114', 'vk5.128', 'vk5.594', 'vk5.1013', 'vk5.1347']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The mirror image K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2U1O3O4U3U4O5U2U5
R3 orbit {'O1O2U1O3O4U3U4O5U2U5'}
R3 orbit length 1
Gauss code of -K O1O2U3O4O3U5U4O5U1U2
Gauss code of K* O1O2U3O4O3U5U4O5U1U2
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 0 -1 1 1],[ 1 0 1 0 0 1],[ 0 -1 0 -1 1 1],[ 1 0 1 0 1 0],[-1 0 -1 -1 0 0],[-1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 -1 -1],[-1 0 0 0 -1],[-1 0 0 -1 0],[ 1 0 1 0 0],[ 1 1 0 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,1,1,0,0,1,1,0,0]
Phi over symmetry [-1,-1,1,1,0,0,1,1,0,0]
Phi of -K [-1,-1,1,1,0,1,2,2,1,0]
Phi of K* [-1,-1,1,1,0,1,2,2,1,0]
Phi of -K* [-1,-1,1,1,0,0,1,1,0,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial -8z-15
Enhanced Jones-Krushkal polynomial -8w^2z-15w
Inner characteristic polynomial t^4+2t^2+1
Outer characteristic polynomial t^5+6t^3+5t
Flat arrow polynomial 4*K1**2 - 2*K2 - 1
2-strand cable arrow polynomial -1536*K1**4 - 1280*K1**2*K2**2 + 1728*K1**2*K2 + 400*K1**2 + 736*K1*K2*K3 - 176*K2**4 + 96*K2**2*K4 - 128*K2**2 - 48*K3**2 - 4*K4**2 + 210
Genus of based matrix 0
Fillings of based matrix [[{1, 5}, {3, 4}, {2}], [{3, 5}, {1, 4}, {2}]]
If K is slice True
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